Exact properties of an integrated correlator in $$ \mathcal{N} $$ = 4 SU(N) SYM

Dorigoni, Daniele; Green, Michael B.; Wen, Congkao

Abstract

We present a novel expression for an integrated correlation function of four superconformal primaries in SU(N) N = 4 supersymmetric Yang-Mills (N = 4 SYM) theory. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. In this paper the correlator is re-expressed as a sum over a two dimensional lattice that is valid for all N and all values of the complex Yang-Mills coupling τ=θ/2π+4πi/g2YM. In this form it is manifestly invariant under SL(2, ℤ) Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the SU(N) correlator to the SU(N + 1) and SU(N − 1) correlators. For any fixed value of N the correlator can be expressed as an infinite series of non-holomorphic Eisenstein series, E(s;τ,τ¯¯¯) with s ∈ ℤ, and rational coefficients that depend on the values of N and s. The perturbative expansion of the integrated correlator is an asymptotic but Borel summable series, in which the n-loop coefficient of order (gYM/π)2n is a rational multiple of ζ(2n + 1). The n = 1 and n = 2 terms agree precisely with results determined directly by integrating the expressions in one-loop and two-loop perturbative N = 4 SYM field theory. Likewise, the charge-k instanton contributions (|k| = 1, 2, . . .) have an asymptotic, but Borel summable, series of perturbative corrections. The large-N expansion of the correlator with fixed τ is a series in powers of N12−ℓ (ℓ ∈ ℤ) with coefficients that are rational sums of E(s;τ,τ¯¯¯) with s ∈ ℤ + 1/2. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider the ’t Hooft topological expansion of large-N Yang-Mills theory in which λ=g2YMN is fixed. The coefficient of each order in the 1/N expansion can be expanded as a series of powers of λ that converges for |λ| < π2. For large λ this becomes an asymptotic series when expanded in powers of 1/λ−−√ with coefficients that are again rational multiples of odd zeta values, in agreement with earlier results and providing new ones. We demonstrate that the large-λ series is not Borel summable, and determine its resurgent non-perturbative completion, which is O(exp(−2λ−−√)).

Citation

Dorigoni, D., Green, M. B., & Wen, C. (2021). Exact properties of an integrated correlator in $$ \mathcal{N} $$ = 4 SU(N) SYM. Journal of High Energy Physics, 2021(5), Article 89. https://doi.org/10.1007/jhep05%282021%29089

• Journal Article Type: Article
• Acceptance Date: Apr 19, 2021
• Online Publication Date: May 12, 2021
• Publication Date: 2021
• Deposit Date: Jun 29, 2021
• Publicly Available Date: Aug 12, 2021
• Journal: Journal of High Energy Physics
• Print ISSN: 1126-6708
• Electronic ISSN: 1029-8479
• Publisher: Scuola Internazionale Superiore di Studi Avanzati (SISSA)
• Peer Reviewed: Peer Reviewed
• Volume: 2021
• Issue: 5
• Article Number: 89
• DOI: https://doi.org/10.1007/jhep05%282021%29089
• Public URL: https://durham-repository.worktribe.com/output/1246267
• Related Public URLs: https://arxiv.org/abs/2102.09537